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Bodieâˆ’Kaneâˆ’Marcus: III. Debt Securities 9. Bond Prices and Yields Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

303

9 Bond Prices and Yields

Inflation in Coupon Principal Total

TA B L E 9.1

Time Year Just Ended Par Value Payment Repayment Payment

Principal and

interest payments 0 $1,000.00

for a Treasury 1 2% 1,020.00 $40.80 0 $ 40.80

Inflation 2 3 1,050.60 42.02 0 42.02

Protected 3 1 1,061.11 42.44 $1,061.11 1,103.55

Security

The real rate of return is precisely the 4% real yield on the bond:

1 Nominal return 1.0608

Real return 1 .04, or 4%

1 Inflation 1.02

One can show in a similar manner (see problem 12 in the end-of-chapter questions) that the

rate of return in each of the three years is 4% as long as the real yield on the bond remains

constant. If real yields do change, then there will be capital gains or losses on the bond. In

mid-2002, the real yield on TIPS bonds was about 2.5%.

9.2 BOND PRICING

Because a bondâ€™s coupon and principal repayments all occur months or years in the future, the

price an investor would be willing to pay for a claim to those payments depends on the value

of dollars to be received in the future compared to dollars in hand today. This â€œpresent valueâ€

calculation depends in turn on market interest rates. As we saw in Chapter 5, the nominal risk-

free interest rate equals the sum of (1) a real risk-free rate of return and (2) a premium above

the real rate to compensate for expected inflation. In addition, because most bonds are not

riskless, the discount rate will embody an additional premium that reflects bond-specific char-

acteristics such as default risk, liquidity, tax attributes, call risk, and so on.

We simplify for now by assuming there is one interest rate that is appropriate for discount-

ing cash flows of any maturity, but we can relax this assumption easily. In practice, there may

be different discount rates for cash flows accruing in different periods. For the time being,

however, we ignore this refinement.

To value a security, we discount its expected cash flows by the appropriate discount rate.

The cash flows from a bond consist of coupon payments until the maturity date plus the final

payment of par value. Therefore

Bond value Present value of coupons Present value of par value

If we call the maturity date T and call the discount rate r, the bond value can be written as

Coupon Par value

T

Bond value (9.1)

(1 r)t (1 r)T

t 1

The summation sign in Equation 9.1 directs us to add the present value of each coupon pay-

ment; each coupon is discounted based on the time until it will be paid. The first term on the

right-hand side of Equation 9.1 is the present value of an annuity. The second term is the pres-

ent value of a single amount, the final payment of the bondâ€™s par value.

You may recall from an introductory finance class that the present value of a $1 annuity

( )

1 1

that lasts for T periods when the interest rate equals r is r 1 (1 r)T . We call this

Bodieâˆ’Kaneâˆ’Marcus: III. Debt Securities 9. Bond Prices and Yields Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

304 Part THREE Debt Securities

1

expression the T-period annuity factor for an interest rate of r.3 Similarly, we call (1 r)T the

PV factor, i.e., the present value of a single payment of $1 to be received in T periods. There-

fore, we can write the price of the bond as

( )

1 1 1

Price Coupon 1 Par value (9.2)

r r)T r)T

(1 (1

Coupon Annuity factor(r, T ) Par value PV factor(r, T )

We discussed earlier an 8% coupon, 30-year maturity bond with par value of $1,000 paying

60 semiannual coupon payments of $40 each. Suppose that the interest rate is 8% annually,

9.2 EXAMPLE or r 4% per six-month period. Then the value of the bond can be written as

60

Bond Pricing $40 $1,000

Price

(1.04)t (1.04)60

t 1

$40 Annuity factor(4%, 60) $1,000 PV factor(4%, 60)

It is easy to confirm that the present value of the bondâ€™s 60 semiannual coupon payments

of $40 each is $904.94, and that the $1,000 final payment of par value has a present value of

$95.06, for a total bond value of $1,000. You can calculate the value directly from Equation

9.2, perform these calculations on any financial calculator,4 or use a set of present value tables.

In this example, the coupon rate equals the market interest rate, and the bond price

equals par value. If the interest rate were not equal to the bondâ€™s coupon rate, the bond

would not sell at par value. For example, if the interest rate were to rise to 10% (5% per six

months), the bondâ€™s price would fall by $189.29, to $810.71, as follows

$40 Annuity factor (5%, 60) $1,000 PV factor(5%, 60)

$757.17 $53.54 $810.71

At a higher interest rate, the present value of the payments to be received by the bondholder

is lower. Therefore, the bond price will fall as market interest rates rise. This illustrates a cru-

cial general rule in bond valuation. When interest rates rise, bond prices must fall because the

present value of the bondâ€™s payments is obtained by discounting at a higher interest rate.

Figure 9.3 shows the price of the 30-year, 8% coupon bond for a range of interest rates in-

cluding 8%, at which the bond sells at par, and 10%, at which it sells for $810.71. The nega-

tive slope illustrates the inverse relationship between prices and yields. Note also from the

figure (and from Table 9.2) that the shape of the curve implies that an increase in the interest

3

Here is a quick derivation of the formula for the present value of an annuity. An annuity lasting T periods can be viewed

as an equivalent to a perpetuity whose first payment comes at the end of the current period less another perpetuity whose

first payment doesnâ€™t come until the end of period T 1. The immediate perpetuity net of the delayed perpetuity pro-

vides exactly T payments. We know that the value of a $1 per period perpetuity is $1/r. Therefore, the present value of

1 1

the delayed perpetuity is $1/r discounted for T additional periods, or . The present value of the annuity is

r (1 r)T

( )

1 1

the presentvalue of the first perpetuity minus the present value of the delayed perpetuity, or 1 .

(1 r)T

r

4

On your financial calculator, you would enter the following inputs: n (number of periods) 60; FV (face or future

value) 1000; PMT (payment each period) 40; i (per period interest rate) 4%; then you would compute the

price of the bond (COMP PV or CPT PV). You should find that the price is $1,000. Actually, most calculators will dis-

play the result as minus $1,000. This is because most (but not all) calculators treat the initial purchase price of the

bond as a cash outflow. We will discuss financial calculators more fully in a few pages.

Bodieâˆ’Kaneâˆ’Marcus: III. Debt Securities 9. Bond Prices and Yields Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

305

9 Bond Prices and Yields

$3,500

$3,000

$2,500

Bond price

$2,000

$1,500

$1,000

$810.71

$500

$0

0% 5% 10% 15% 20%

Interest rate

F I G U R E 9.3

The inverse relationship between bond prices and yields. Price of an 8% coupon bond with 30-year maturity making

semiannual coupon payments.

Bond Price at Given Market Interest Rate

TA B L E 9.2

Time to Maturity 4% 6% 8% 10% 12%

Bond prices

at different

1 year $1,038.83 $1,019.13 $1,000.00 $981.41 $963.33

interest rates (8%

10 years 1,327.03 1,148.77 1,000.00 875.38 770.60

coupon bond,

20 years 1,547.11 1,231.15 1,000.00 828.41 699.07

coupons paid

30 years 1,695.22 1,276.76 1,000.00 810.71 676.77

semiannually)

rate results in a price decline that is smaller than the price gain resulting from a decrease of

equal magnitude in the interest rate. This property of bond prices is called convexity because

of the convex shape of the bond price curve. This curvature reflects the fact that progressive

increases in the interest rate result in progressively smaller reductions in the bond price.5

Therefore, the price curve becomes flatter at higher interest rates. We will return to the issue

of convexity in the next chapter.

5

The progressively smaller impact of interest rate increases results from the fact that at higher rates the bond is worth

less. Therefore, an additional increase in rates operates on a smaller initial base, resulting in a smaller price reduction.

Bodieâˆ’Kaneâˆ’Marcus: III. Debt Securities 9. Bond Prices and Yields Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

306 Part THREE Debt Securities

>

2. Calculate the price of the bond for a market interest rate of 3% per half year. Com-

Concept

pare the capital gains for the interest rate decline to the losses incurred when the

CHECK rate increases to 5%.

Corporate bonds typically are issued at par value. This means the underwriters of the bond

issue (the firms that market the bonds to the public for the issuing corporation) must choose a

coupon rate that very closely approximates market yields. In a primary issue of bonds, the

underwriters attempt to sell the newly issued bonds directly to their customers. If the coupon

rate is inadequate, investors will not pay par value for the bonds.

After the bonds are issued, bondholders may buy or sell bonds in secondary markets, such

as the one operated by the New York Stock Exchange or the over-the-counter market, where

most bonds trade. In these secondary markets, bond prices move in accordance with market

forces. The bond prices fluctuate inversely with the market interest rate.

The inverse relationship between price and yield is a central feature of fixed-income secu-

rities. Interest rate fluctuations represent the main source of risk in the bond market, and we

devote considerable attention in the next chapter to assessing the sensitivity of bond prices to

market yields. For now, however, it is sufficient to highlight one key factor that determines

that sensitivity, namely, the maturity of the bond.

A general rule in evaluating bond price risk is that, keeping all other factors the same, the

longer the maturity of the bond, the greater the sensitivity of its price to fluctuations in the in-

terest rate. For example, consider Table 9.2, which presents the price of an 8% coupon bond

at different market yields and times to maturity. For any departure of the interest rate from 8%

(the rate at which the bond sells at par value), the change in the bond price is smaller for

shorter times to maturity.

This makes sense. If you buy the bond at par with an 8% coupon rate, and market rates sub-

sequently rise, then you suffer a loss: You have tied up your money earning 8% when alterna-

tive investments offer higher returns. This is reflected in a capital loss on the bondâ€”a fall in

its market price. The longer the period for which your money is tied up, the greater the loss

and, correspondingly, the greater the drop in the bond price. In Table 9.2, the row for one-year

maturity bonds shows little price sensitivityâ€”that is, with only one yearâ€™s earnings at stake,

changes in interest rates are not too threatening. But for 30-year maturity bonds, interest rate

swings have a large impact on bond prices.

This is why short-term Treasury securities such as T-bills are considered to be the safest.

They are free not only of default risk but also largely of price risk attributable to interest rate

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